i may be wrong but, for a very long time, mental health was really a taboo topic, especially in korea. things have been changing for the last couple of years (like +10y) so maybe thats why he doesn't bring this up to her - although i, too, wish to see a more deep development of this issue and, of course it doesn't excuse the way he tells her to "get over it" etc.
maybe his way of think comes from this old viewing on mental health (but this is just my speciation) - or he can just be an asshole (he kinda is in someway, based on some flashback about the breakup, which i'm dying to know the full context) and has this shallow view that it can be solved with pure willpower - and maybe this too will change after this cp where she does tru to end her life (or so it seems)
or even, maybe this is just the kind of story for the sake of the plot, like if she actually goes to the doctor, maybe they would not have that much baggage for the present story (and i would be a little disappointed if that was the case but not surprised)

Yes he's hot,tall but takes good care of her?? I thought so too before but now I'm sure they both need professional help,her way more then him obviously. The most trauma that guy went through is, correct me if I'm wrong, getting blamed for hurting someone during a sport where people get injured often. Meanwhile she lost everything,she lived a very difficult life from childhood and when she was finally escaping that life, her house burned along with her mother whom she found literally burning. Telling her to get over it is such a sick thing to say and even worse when so little time has passed. Only 6 months and she's supposed to get over it?? She's so much not getting over it that her mind is erasing it's memories so it doesn't suffer. I hope to see some development in this guy's personality or at least give some valid reasons as to why he didn't bring her to a psychiatrist.